Comparisons and asymptotics for empty space hazard functions of germ-grain models

We study stochastic properties of the empty space for stationary germ-grain models in $\R^d$, in particular we deal with the inner radius of the empty space with respect to a general structuring element which is allowed to be lower-dimensional. We consider Poisson cluster germ-grain models and Boolean models with grains that are clusters of convex bodies and show that more variable size of clusters results in stochastically greater empty space in terms of the empty space hazard function. We also study impact of clusters being more spread in the space on the value of the empty space hazard. Further we obtain asymptotic behavior of the empty space hazard functions at zero and at infinity.